Sorbent materials are widely used for the pumping, storage, and purification of gases and gas mixtures and as elements of refrigeration systems. These solid sorbent materials are contained in a pressure vessel that includes one or more apertures through which gas is added or removed, and free space or ullage. Since the quantity of adsorbed gas is proportional to the amount of sorbent, storage capacity is maximized when the sorbent completely fills the vessel and the ullage is zero. However, gas flow into and out of porous materials is slower than free convection through the ullage, so tanks without ullage may not provide adequate uptake and discharge rates for specific applications.
Transport of gas into and out of the sorbent couples flow through the apertures and ullage by free convection to flow through the sorbent by porous convection. The former is described by the Navier-Stokes equations of fluid dynamics, while the latter is described by D'Arcy's equation and its variants. Convection, which is driven by pressure gradients, is always faster through free space than through porous media for the same geometry and pressure drop.
The total rate of gas flow into or out of the sorbent is proportional the surface area that is exposed to ullage, while the total capacity is proportional the sorbent's volume. A standard method for increasing the surface area, S, in a fixed total volume, V, is to pack the pressure vessel with spherical particles whose surface area to volume ratio is inversely proportional to the particle radius r:
      S    V    =            3      r        .  A problem with this approach of using powders or granules is revealed by Kepler's conjecture, which states that the maximum packing density for monodisperse spheres is
            π              18              ~    0.74    .Choosing spherical particles with two discrete but different radii increases i the maximum achievable packing density to 0.82, [D. de Laat, F. M. de Oliveira Filho, and F. Vallentin, “Upper Bounds for Packings of Spheres of Several Radii”, arXiv:1206.2608v1, 12 Jun. 2012] but complicates fabrication of a sorbent array because ordered placement of spheres with different diameters is required to achieve this maximum. There exists a need for improved sorbents, systems, and methods of storing, pumping, and purifying gases.